An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Cocycles and the structure of ergodic group actions

We show, for a large class of groups, the existence of cocycles taking values in these groups and which define ergodic skew products. We apply this to prove a generalization of Ambrose’s representation theorem for ergodic actions of these groups.     

An integral formula on the Heisenberg group

Let $\mathbb{H }^n$ denote the $(2n+1)$ -dimensional (sub-Riemannian) Heisenberg group. In this note, we shall prove an integral identity (see Theorem 1.2) that generalizes a formula obtained in the Seventies by Reilly (Indiana Univ Math J 26(3):459–472, 1977). Some first applications will be given in Sect. 4.     

Multitime dynamic programming for multiple integral actions

This paper introduces a new type of dynamic programming PDE for optimal control problems with performance criteria involving multiple integrals. The main novel feature of the multitime dynamic programming PDE, relative to the standard Hamilton-Jacobi-Bellman PDE, is that it is connected to the multitime maximum principle and is of divergence type. Introducing a generating vector field for the maximum value function, we present an interesting and useful connection between the multitime maximum principle and the multitime dynamic programming, characterizing the optimal control by means of a multitime Hamilton-Jacobi-Bellman (divergence) PDE that may be viewed as a feedback law. Section 1 recalls the multitime maximum principle. Section 2 shows how a multitime control dynamics determines the multitime Hamilton-Jacobi-Bellman PDE via a generating vector field of the value function. Section 3 gives an example of two-time dynamics with nine velocities proving that our theory works well. Section 4 shows that the Hamilton PDEs are characteristic PDEs of multitime Hamilton-Jacobi PDE and that the costates in the multitime maximum principle are in fact gradients of the components of the generating vector field.     

Non-treeability for product group actions

A product of two lcsc non-compact groups G ₁ × G ₂ acting freely by measure preserving transformation on a standard Borel probability space gives rise to a non-treeable equivalence relation unless both groups are amenable. Partially supported by NSF grant DMS 0140503     

Generators for amenable group actions

A generalisation of Krieger's finite generator theorem is proved for free actions of countable amenable groups on a non-atomic Lebesgue probability space.     

An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the -class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture [Yongbin Ruan, The cohomology ring of crepant resolutions of orbifolds, in: Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117-126].     

Equivariant autoequivalences for finite group actions

The familiar Fourier-Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut(DbG(X)) with the group of autoequivalences of DG(X). We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.     

Entropy rigidity for semisimple group actions

We show for the standard actions of higher rank semisimple Lie groups and their discrete subgroups on compact manifolds that the entropy is invariant under small perturbations.     

Exact cohomology sequences with integral coefficients for torus actions

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces X, with free equivariant cohomology over Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.     

Fuzzy relation spaces for group decision theory: An application

In a previous issue the authors proposed a fuzzy relation-theoretic model for group decision theory, and proved a number of theorems concerning its structural properties. The present paper, a revision of a 1977 IEEE CDC Proceedings talk, exemplifies one possible application of the theory; updates references to other published applications and closely related work; and expands suggestions for potential research along several lines of theory and applications.     

Integrable systems and group actions

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.     

Hierarchical forecasting based on AR-GARCH model in a coherent structure

This paper compares the accuracy of the aggregate forecasting with the bottom-up forecasting based on AR-GARCH model for the return rate of simulated Dow Jones Industrial Average. Most of the existing stock price index studies did not consider the hierarchical structure and often missed the coherent relationships between individual components. In this experiment, we simulated 30 coherent components based on AR(2)-GARCH(1, 1) model. Then we evaluated the performance of both forecasting methods ignoring the coherent structure. The results of our experiment indicated that the accuracy of forecasting method varied depending on the correlation degree of 30 coherent components, however the data noise did not significantly influenced the performance of hierarchical forecasting method.     

Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G  . We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group _A5$A_5$, and give various other examples.     

Polish group actions and effectivity

We extend a theorem of Barwise and Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.